The term laser is an anachronism for Light
Amplification by Stimulated
Emission of Radiation.
The use of each one of those words will be explained win the text development
below.

Light is represented by a wave that is the
superposition of various (or just one) waves of the form
. In the previous
expression, is the
amplitude of the wave,
is the wave vector
defined by with
where
is the wavelength
of the perturbation. The parameter
represent the angular frequency of the wave. Here,
is the frequency of the
oscillation (reference to waves). Finally,
the parameter is an arbitrary
phase that is determined by the value of the wave at
and
.

The graph above represents a snap shot of a one-dimensional wave,
, let us say, at . In that case,
the phase can be written in terms of the coordinate position
and, in general,
.

Coherent and Incoherent Light

Ordinary light is made out of the
superposition of various waves, as the one presented above, with
different wavelength and/or frequency and phases. The resulting beam is
say to be non-monochromatic
incoherent light.

If a color filter is placed in the
path of such a beam of light, the result is
Monochromatic Incoherent Light. A color filter will select a very
small range of frequencies (colors) for transmission through the filter;
all the other frequencies are blocked by the filter.Incoherent light refers to the superposition of light with different
phases.

The light emitted by a laser is
not only monochromatic but also is coherent.Coherent light refers to the superposition
of waves with the same frequency, wave vector, and phase. In three
dimensions, the wave vector indicates the direction of propagation of
the wave. In the case of coherent light, it is not just required that
the wavelength of the different components of the addition are the same
(associated to the magnitude of the wave vector) but also the direction
of propagation of these components must be the same (non-diverging
beams).

The upper graph represents waves with different wavelength (longer
for red, smaller for violet). The
lower picture has the same characteristic as the upper picture but
with additional arbitrary phases among the different rays.

The top wave on the
graph above represents the addition (superposition) of all the waves
below in the graph. The waves below are monochromatic (all have the same
frequency) but they have different phases. The addition of the waves has
been graphed at the same scale as the other. Notices that rather than a
constructive addition of these waves, we have a some how destructive
superposition of these waves with a resulting amplitude smaller than the
amplitude of the component waves.

The top wave on the
graph above represents the addition (superposition) of all the waves
below in the graph. The waves below are monochromatic (all have the same
frequency) but, in this case, they have the same phase. The addition of
the waves has been graphed at the same scale as the components of the
addition. Notice that now the superposition is constructive and the
amplitude of the wave is much bigger than the individual components.

Thus the characteristic of a laser are

light is nearly monochromatic, same
,

light is coherent; that is, waves are exactly in
phase, same ,

light does not diverges, same
; and,

light is very intense, the resulting amplitude, ,
is very high.

The laser effect is produced in a media when more atoms in the media
are in an excited state than in the ground state of the atom and the
electrons in the excited state jump back to the ground state emitting
photons with the characteristics expressed above. This is called
population inversion. How this is achieved
will be presented below.

Three Levels Laser

The simples operational laser is the three levels laser that involved
three atomic levels to produce the laser effect. These levels are the
ground state of the atom, an regular excited state, and a meta-stable
state of the atom.

Regular excited states are atomic levels where an excited electron
can stay after absorbing energy. However, the time that the electron
last in such state is of the orders of 10-8 seconds. On the
other side, a meta-stable state is an excited state where the electron
can last as long as five orders of magnitude longer, 10-3
seconds. Such a long lasting excited state can result in the media
reaching the status of population inversion where more atoms are in the
excited state than in the ground state.

The sequence of drawings presented at the left
corresponds to an atom going through the atomic process involved in the
functioning of a laser. The first drawing of the
sequence shows the pumping of energy in the atom by incoherent radiation photons
while the electron
is in the ground state, .
Only one photon from the incoherent radiation is absorbed by the
electron pumping it to the regular excited state, .
The electron stays on the excited time for a time of the order of 10-8
seconds,
.
The electron emits a photon with an energy equals to the difference in
energy between the regular excited state and the meta-stable state.
While emitting, the electron jumps from the regular excited state to the
meta-stable state, .
Now, the electron stays in the meta-stable state for a time 105
times longer than when it was in the regular excited state. Thus, the
electron stays in the meta-state for about 10-3 seconds,
.
The last drawing of the sequence shows the emission of the laser photon.
This photon has an energy corresponding to the transition energy from
the meta-stable state to the ground state of the atom
.

In a laser there are three basic processes, stimulated absorption,
spontaneous emission, and stimulated emission. The drawings below shows
these processes.

Stimulated
Absorption: In this process a photon with an energy equals to the
difference in energy between two of the atomic levels interacts with the
atom. The electron of the atom is in the lower energy state, absorbs the
photon, and jumps to the higher energy state (excited state). The photon
is completely absorbed by the atom. The final state of the atom is
excited.

Spontaneous
emission: In this case, one of the electron is not in the lowest
possible energy state; thus, the atom is excited. Spontaneously, the
atom emits a photon with an energy equals to the difference in energy
between the energy when the electron is in the excite state and the
energy of the level when the electron is in the relaxed state (lower
energy state). During this emission the electron jumps down to the lower
energy state. The atoms give up energy which is carried by the emitted
photon and it is left unexcited. On the other side, the emitted photon
has an energy equals to the difference in energy between the two atomic
states involved. However, there is not a prefer direction of motion for
the emitted photon; in fact, the direction of emission is completely
random.

Stimulated emission: When a photon with the
same energy as the difference between two atomic levels reaches the atom
and the corresponding electron to those levels is in the excited state,
the electron jumps down to the lower energy state emitting an additional
photon. After this process, two photons are emerging from the atom for
every incident photon that interacts with the atom inducing the
emission.

Emission and Absorption of Photons

In this section, the relations between the atomic parameters and the
incident electromagnetic radiation are study. Relations between the
parameters associated with the stimulated absorption, spontaneous
emission, and stimulated emission are presented.

The drawing on the left represents the
two energy levels involved in the emission and absorption of photons.
With the lower energy and
the higher energy level. The
sample to be study has many atoms with exactly the same characteristics.
In addition, the sample has many atoms with the electron on the higher energy state and
many atoms with the electron in the lower energy state. All these
atoms are in thermal equilibrium at the absolute temperature
with the electromagnetic
radiation of frequency
and energy density
.
Let us call the number of atoms
in the state
and
the number of atoms in the state
.
Thus, the number of atoms that absorb a photon and jump from the state
with energy to the state with
energy
can be written as

where
is a constant that
depends on the properties of the states
and
.

In the same form, the number of atoms that can spontaneously
drop from the state
to the state
by emitting a photon is
where is a constant that
represents the probability for this atomic transition and the corresponding photon
emission.

At the same time,
if
is the corresponding
constant representing the probability
for stimulated emission, the number of atoms that emit photons by
either spontaneous or stimulated emission can be written as
.

For the atomic system media at thermal equilibrium with the
radiation, the number of
transitions from to
is the same as the number of
transitions from
to ,

At the previous equality, the radiation
energy density can be obtained,

Applying, Maxwell-Boltzmann distribution to the number of
atoms in a particular state,
and where
is Boltzmann constant. The ratio between
the two number of atoms in the two states is
. Therefore,

The previous result can be compared with Planck radiation
formula, , and the following
equalities among the various quantities are deduced,

and

These result imply the following conclusions:

:

Stimulated emission DOES occur.

The stimulated emission and the stimulated
absorption have the same weight in the transition numbers. That
is, if the number of atoms in the two atomic states are about
the same, the probability for an atom of the simple to undergo
stimulate d emission is the same as the probability of an atom
in the simple to undergo and stimulated absorption.

An incident photon of energy
has the same probability
of causing an atom to emit a photon of this energy as of being
absorbed by an atom of the simple when the radiation and the
simple are at thermal equilibrium.

:

The spontaneous emission occurrence in the
media increases rapidly with the energy difference between the
two atomic states when compared to the stimulated emission.